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SIAM Journal on Computing
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SIAM Journal on Computing
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ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
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Let Gbe an unweighted and undirected graph of nnodes, and let Dbe the n×nmatrix storing the All-Pairs-Shortest-Path distances in G. Since Dcontains integers in [n] 茂戮驴 + 茂戮驴, its plain storage takes n2log(n+ 1) bits. However, a simple counting argument shows that (n2茂戮驴 n)/2 bits are necessary to store D. In this paper we investigate the question of finding a succinct representation of Dthat requires O(n2) bits of storage and still supports constant-time access to each of its entries. This is asymptotically optimal in the worst case, and far from the information-theoretic lower-bound by a multiplicative factor log23 茂戮驴 1.585. As a result O(1) bits per pairs of nodes in Gare enough to retain constant-time access to their shortest-path distance. We achieve this result by reducing the storage of Dto the succinct storage of labeled trees and ternary sequences, for which we properly adapt and orchestrate the use of known compressed data structures.